Categories: Quantum information.

# Quantum gate

In quantum computing, **quantum gates** are the equivalent
of classical binary logic gates such as $\mathrm{NOT}$, $\mathrm{AND}$, etc.
Because of the continuous nature of qubits,
the number of possible quantum gates is uncountably infinite,
so we only consider the most important examples here.

## One-qubit gates

As an example, consider the following most general single-qubit state $\ket{\psi}$:

$\begin{aligned} \ket{\psi} = \alpha \ket{0} + \beta \ket{1} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \end{aligned}$Arguably the most famous and most fundamental quantum gates are the **Pauli matrices**:

They have the following effect on $\ket{\psi}$.
Note that $X$ is equivalent to the classical $\mathrm{NOT}$ gate
(and is often given that name),
and $Z$ is sometimes called the **phase-flip gate**:

In fact, $Z$ is a specific case of the **phase shift gate** $R_\phi$,
which modifies the qubit’s phase without changing its amplitudes.
For an angle $\phi$, it is given by:

For $\phi = \pi$, we recover the Pauli-$Z$ gate. In general, the action of $R_\phi$ is as follows:

$\begin{aligned} R_\phi \ket{\psi} = \begin{bmatrix} \alpha \\ e^{i \phi} \beta \end{bmatrix} \end{aligned}$Two common special cases of $R_\phi$ are $\phi = \pi/2$ and $\phi = \pi/4$, respectively called $S$ and $T$:

$\begin{aligned} \boxed{ S = R_{\pi/2} = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} } \qquad \quad \boxed{ T = R_{\pi/4} = \frac{1}{\sqrt{2}} \begin{bmatrix} \sqrt{2} & 0 \\ 0 & 1 + i \end{bmatrix} } \end{aligned}$Finally, we have the **Hadamard gate** $H$,
which is defined as follows:

Its action consists of rotating the qubit by $\pi$ around the axis $(X + Z) / \sqrt{2}$ of the Bloch sphere:

$\begin{aligned} H \ket{\psi} = \frac{1}{\sqrt{2}} \begin{bmatrix} \alpha + \beta \\ \alpha - \beta \end{bmatrix} \end{aligned}$Notably, it maps the eigenstates of $X$ and $Z$ to each other, and is its own inverse (i.e. unitary):

$\begin{aligned} H \ket{0} = \ket{+} \qquad H \ket{1} = \ket{-} \qquad H \ket{+} = \ket{0} \qquad H \ket{-} = \ket{1} \end{aligned}$The **Clifford gates** are a set including $X$, $Y$, $Z$, $H$ and $S$,
or more generally any gates that rotate
by multiples of $\pi/2$ around the Bloch sphere.
This set is **not universal**, meaning that if we start from $\ket{0}$,
we can only reach $\ket{0}$, $\ket{1}$, $\ket{+}$, $\ket{-}$, $\ket{+i}$ $\ket{-i}$ using these gates.

If we add *any* non-Clifford gate, for example $T$,
then we can reach any point on the Bloch sphere,
which means that the set is **universal**.

However, there is a problem: a qubit has an uncountable infinity of states,
but a quantum circuit consists of a countably infinite sequence of gates, at most.
Therefore, technically, we can never reach the whole Bloch sphere,
but we *can* come up with circuits that approximate a target state to some degree $\varepsilon$.
This is the definition of universality:
any state can be approximated.

## Two-qubit gates

As an example, let us consider the following two pure one-qubit states $\ket{\psi_1}$ and $\ket{\psi_2}$:

$\begin{aligned} \ket{\psi_1} = \alpha_1 \ket{0} + \beta_1 \ket{1} = \begin{bmatrix} \alpha_1 \\ \beta_1 \end{bmatrix} \qquad \quad \ket{\psi_2} = \alpha_2 \ket{0} + \beta_2 \ket{1} = \begin{bmatrix} \alpha_2 \\ \beta_2 \end{bmatrix} \end{aligned}$The composite state of both qubits, assuming they are pure, is then their tensor product $\otimes$:

$\begin{aligned} \ket{\psi_1 \psi_2} = \ket{\psi_1} \otimes \ket{\psi_2} &= \alpha_1 \alpha_2 \ket{00} + \alpha_1 \beta_2 \ket{01} + \beta_1 \alpha_2 \ket{10} + \beta_1 \beta_2 \ket{11} \\ &= c_{00} \ket{00} + c_{01} \ket{01} + c_{10} \ket{10} + c_{11} \ket{11} \end{aligned}$Note that a two-qubit system may be entangled, in which case the coefficients $c_{00}$ etc. cannot be written as products, i.e. $\ket{\psi_2}$ cannot be expressed separately from $\ket{\psi_1}$, and vice versa. In other words, the action of a two-qubit gate can be expressed in the basis of $\ket{00}$, $\ket{01}$, $\ket{10}$ and $\ket{11}$, but not always in the basis of $\ket{0}_1$, $\ket{1}_1$, $\ket{0}_2$ and $\ket{1}_2$.

With this noted, the first two-qubit gate is $\mathrm{SWAP}$, which simply swaps $\ket{\psi_1}$ and $\ket{\psi_2}$:

$\begin{aligned} \boxed{ \mathrm{SWAP} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} } \end{aligned}$This matrix is given in the basis of $\ket{00}$, $\ket{01}$, $\ket{10}$ and $\ket{11}$. Note that $\mathrm{SWAP}$ cannot generate entanglement, so if its input is separable, its output is too. In any case, its effect is clear:

$\begin{aligned} \mathrm{SWAP} \ket{\psi_1 \psi_2} &= c_{00} \ket{00} + c_{10} \ket{01} + c_{01} \ket{10} + c_{11} \ket{11} \end{aligned}$Next, there is the **controlled NOT gate** $\mathrm{CNOT}$,
which “flips” (applies $X$ to) $\ket{\psi_2}$ if $\ket{\psi_1}$ is true:

That is, it swaps the last two coefficients $c_{10}$ and $c_{11}$ in the composite state vector:

$\begin{aligned} \mathrm{CNOT} \ket{\psi_1 \psi_2} &= c_{00} \ket{00} + c_{01} \ket{01} + c_{11} \ket{10} + c_{10} \ket{11} \end{aligned}$More generally, from every one-qubit gate $U$,
we can define a two-qubit **controlled U gate** $\mathrm{CU}$,
which applies $U$ to $\ket{\psi_2}$ if $\ket{\psi_1}$ is true:

Where the lower-right 2x2 block is simply $U$. The general action of this gate is given by:

$\begin{aligned} \mathrm{CU} \ket{\psi_1 \psi_2} &= c_{00} \ket{00} + c_{01} \ket{01} + (c_{10} u_{00} + c_{11} u_{01}) \ket{10} + (c_{10} u_{10} + c_{11} u_{11}) \ket{11} \end{aligned}$A set of gates is **universal** if all possible mappings
from $n$ to $n$ qubits can be approximated using only these gates.
A minimal universal set is $\{\mathrm{CNOT}, T, S\}$,
and there exist many others.

## References

- J.S. Neergaard-Nielsen,
*Quantum information: lectures notes*, 2021, unpublished. - S. Aaronson,
*Introduction to quantum information science: lecture notes*, 2018, unpublished.